Integrability and non-existence of periodic orbits of a class of Kolmogorov system

Abstract

In this article, we study the integrability and the non-existence of periodic orbits for the planar Kolmogorov differential systems of the form

\dot{x}=x(R_{n-1}(x,y)+P_n(x,y)+S_{n+1}(x,y)),

\dot{y}=y(R_{n-1}(x,y)+Q_n(x,y)+S_{n+1}(x,y))

where n is a positive integer, R_{n-1}, P_n, Q_n and S_{n+1} are homogeneous polynomials of degree n-1, n, n and n+1, respectively. Applications of Kolmogorov systems can be found particularly in modeling population dynamics in biology and ecology.